# math

posted by on .

A certain football league has the following scoring system:

- each field goal counts for 5 points
- each touchdown counts for 3 points

The only way to score points is with some combination of field goals
and touchdowns.

They have noticed that not every score is possible in this league
(for example 1, 2, 4...); in fact they think that they know the
highest score that is impossible to make. I have figured out that this
is 7 points, but I would like to know why. How can I prove that this
score is impossible and that all higher scores are possible?

Generally when two integers, say p and q, are relatively prime (have no common divisor) then there exists integers u and v such that for "any" integer n
n = up + vq
Here p and q are 5 and 3. If the numbers are not relatively prime then the only numbers that can be written like this are all multiples of the gcd of the two numbers.
I know there is some rule that after a certain number all of the positive numbers can be written using only positive integers, but I don't recall it offhand. It should be in an elementary number theory text somewhere; I know I've seen it in several places, but I can't recall it right now.